There are far more effective methods of attacking than the "two-thief" and amplifier-based attacks. I won't discuss them since I don't want them to spread, but you can probably work them out pretty easily on your own.
Possibly, especially if the car manufacturer was clueless and didn't use multiple challenge/response transactions or one time codes, facilitating various possible replay attacks. You don't need to worry about "them spreading", the attacks are published - they have been at Defcon and BlackHat recently, I believe. Anyhow, the info is widely available for anyone who knows how to use Google.
Any reason you don't wrap the key in tin foil?
Well, you could do that and it *might* work (it's actually much less effective than you'd think), but it's still very inconvenient because you can't turn the protection off/on easily. In other words it ruins the convenience of the RKE system.
What you don't realize is that that jammer you are proposing would have to be very indiscriminate - the moment it detects something on one of the assigned frequencies (these systems usually use frequency hopping, not a single fixed frequency), it would have to start jamming right away - all the while possibly interfering with someone else's key fob, garage door openers, alarm remote, etc.
And if it wanted to minimize this and waited until it could positively identify the signal as coming from the key, it would likely be too late - the thief could have recorded the signal already and your subsequent jamming would be useless.
In this context the tin foil would actually be a lot more effective than what you are proposing.
Anyhow, you don't have to have this option active on your car or just don't buy a car with it. A good theft insurance on a new car would go a long way towards mitigating any of these issues. Also, not sure how it is in the US, but in Central/Eastern Europe where car theft is a common problem, it has been a common wisdom for a long time to not rely only on the manufacturer's lock and immobilizer, because the thieves are familiar with them and most have publicly known more or less trivial bypasses. Actually some insurance companies used to turn you down unless you have bought an aftermarket alarm.
X-Y problem, indeed.